Signature approach for pricing and hedging path-dependent options with frictions

TL;DR

This work introduces a path-signature framework to price and hedge path-dependent options in markets with both temporary and permanent frictions, transforming a non-Markovian stochastic control problem into a tractable, Markovian-like system on the extended signature space. It establishes a rigorous verification approach via an infinite-dimensional Riccati equation on the extended tensor algebra, yielding a feedback hedge that is a (potentially infinite) linear combination of time-augmented signature elements. The paper provides existence/uniqueness results, two explicit solvable cases, and numerical demonstrations showing that signature-based strategies remain accurate and robust in frictional markets where market impact smooths trading paths; it also shows how non-signature payoffs can be handled through universal approximation by signature payoffs. Overall, the framework offers a flexible, scalable method to manage path-dependent payoffs under frictions, bridging non-Markovian stochastic control with Riccati-type dynamics. The numerical results underscore the practical viability and advantages of incorporating permanent impact into hedging decisions, especially for complex payoffs.

Abstract

We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an inherently nonlinear and non-Markovian stochastic control problem into a tractable form, yielding hedging strategies in (possibly infinite) linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by non-standard infinite-dimensional Riccati equations on the extended tensor algebra. Numerical experiments demonstrate the effectiveness of these signature-based strategies for pricing and hedging general path-dependent payoffs in the presence of frictions. In particular, market impact naturally smooths optimal trading strategies, making low-truncated signature approximations highly accurate and robust in frictional markets, contrary to the frictionless case.

Figures (6)

• Figure 1: Sample path of trading speed for the European quadratic payoff: explicit expression given by Proposition \ref{['prop: explicit_sol_eu_quad']} vs numerical resolution of Riccati equation with truncated shuffle product. Parameters: $M=2$, $K=S_0$ and $N=200$.
• Figure 2: Sample path of trading speed and inventory for Asian quadratic payoff. Solid line: explicit hedging strategy assuming $\nu=0$ given by Proposition \ref{['prop: explicit_sol_without_permanent_impact']}; dashed lines: signature-based strategies obtained by numerically solving the Riccati equations with projected shuffle product. Parameters: $M=4$, $K=S_0$, and $N=200$.
• Figure 3: Histogram of $R_T^\theta$ for Asian quadratic payoff with $\nu>0$: signature-based strategy vs explicit hedging strategy assuming $\nu=0$ given by Proposition \ref{['prop: explicit_sol_without_permanent_impact']}. Other parameters: $M=4$, $K=S_0$, and $N=200$. $MQV(\theta)=\mathbb{E}\left(R_T^\theta-\frac{\lambda}{2}[R^\theta,R^\theta]_T\right)$.
• Figure 4: Sample paths of inventory $X_t^\theta$ for European call payoff with $\eta=\nu=0$: signature-based approximated strategy vs Bachelier Delta hedging strategy. Parameters: $M=5$, $K=S_0$, and $N=200$.
• Figure 5: Sample paths of trading speed $\theta_t$ and inventory $X_t^\theta$ for European call payoff with $\eta=\nu=0.001$: signature-based approximated strategy vs optimal hedging strategy by solving the HJB equation \ref{['eq: HJB_european_payoff']}. Parameters: $M=5$, $K=S_0$, and $N=200$.

Theorems & Definitions (41)

• Definition 3.1: Shuffle product
• Definition 3.2
• Proposition 3.3: Shuffle property
• Theorem 3.4: Itô's formula
• proof
• Example 4.1
• Theorem 4.2
• proof
• Lemma 4.3
• proof